In my unlikely role as French Strategist and Deep Thinker (you don't suppose there is a language requirement? Non? C'est bien!), I would offer the following motivations:

1. Better to reign in hell than serve in heaven. NATO is a military alliance with many new entrants that do not remember Germany fondly from WWII, and do not remember France as the nation that liberated them from the Soviet Empire. France can influence NATO, but never lead it.

2. Re-shuffle the deck: The US has blessed the creation of the Rapid Reaction Force, provided it is not simply a shuffling of NATO assets. Hmm, new troops, new equipment, new money? No way. If the RRF is vital to creating a European military capability that can give them their own voice, then ending NATO may be the cheapest way to create it. France will be one of the big fish in this puddle.

3. Let's talk Turkey: Turkey will never be admitted to the EU, which has no desire to add to its unassimilated Muslim population. Still, Turkey keeps offering the tired arguemnt that they have been a loyal member of NATO for fifty years. Well, let's end NATO and end this discussion.

That's it - three strikes, and we're out! Ooops, sorry, that is very American.